Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,11,14,12,15,16,13)(17,18)(19,25)(20,29)(21,23)(22,27)(24,26)(28,31)(30,32);(2,5)(6,7)(10,13)(11,16)(14,15)(17,19,26,32,21,28,20,27)(18,22,29,31,23,30,24,25);(1,3,4,8)(2,6,5,7)(18,23)(22,30)(24,29)(25,31)Polycyclic presentation:
2^8|a^2=f,b^2=defh,c^2=eg,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98089, #98091, #98097, … |
| maximal normal subgroup of | #98089, #98091, #98097, … |
| maximal quotient of | #102394, #102418, #109191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2887);